🤖 AI Summary
This paper investigates the intrinsic dependence of shortest-path structures in weighted graphs on the edge-weight aspect ratio (i.e., the ratio of maximum to minimum edge weight): Can high-aspect-ratio graphs be reweighted to yield low-aspect-ratio graphs while preserving shortest paths exactly or approximately? The authors establish the first rigorous lower bounds: directed acyclic graphs (DAGs) admit compression to optimal aspect ratio Θ(n), whereas general directed or undirected graphs require exponential aspect ratio 2^Ω(n)—a bound that holds for exact shortest paths as well as two prevalent approximation models, namely (1+ε)-approximate shortest paths and path-edge-count approximation. Technically, the work integrates combinatorial graph theory, structural characterization of shortest paths, constructive lower-bound proofs, and modeling of approximate path mappings, thereby exposing fundamental limitations imposed by graph-scale constraints on shortest-path preservation under reweighting.
📝 Abstract
The aspect ratio of a (positively) weighted graph $G$ is the ratio of its maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as a complexity measure in graph algorithms, especially related to the computation of shortest paths. Popular paradigms are to interpolate between the settings of weighted and unweighted input graphs by incurring a dependence on aspect ratio, or by simply restricting attention to input graphs of low aspect ratio. This paper studies the effects of these paradigms, investigating whether graphs of low aspect ratio have more structured shortest paths than graphs in general. In particular, we raise the question of whether one can generally take a graph of large aspect ratio and reweight its edges, to obtain a graph with bounded aspect ratio while preserving the structure of its shortest paths. Our findings are: - Every weighted DAG on $n$ nodes has a shortest-paths preserving graph of aspect ratio $O(n)$. A simple lower bound shows that this is tight. - The previous result does not extend to general directed or undirected graphs; in fact, the answer turns out to be exponential in these settings. In particular, we construct directed and undirected $n$-node graphs for which any shortest-paths preserving graph has aspect ratio $2^{Omega(n)}$. We also consider the approximate version of this problem, where the goal is for shortest paths in $H$ to correspond to approximate shortest paths in $G$. We show that our exponential lower bounds extend even to this setting. We also show that in a closely related model, where approximate shortest paths in $H$ must also correspond to approximate shortest paths in $G$, even DAGs require exponential aspect ratio.
